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| [[File:Harmoniki.png|thumb|300px|right|Cubic harmonics]]
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| In fields like [[computational chemistry]] and [[Solid-state physics|solid-state]] and [[Condensed matter physics|condensed matter]] physics the so-called [[atomic orbitals]], or [[spin-orbital]]s, as they appear in textbooks<ref name=Messiah>
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| {{cite book |author=Albert Messiah |title=Quantum Mechanics |year= 1999 |publisher= Dover Publications
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| |url=http://books.google.com/books?id=mwssSDXzkNcC |isbn=0-486-40924-4 }}
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| </ref><ref name=Gasiorowicz>
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| {{cite book |author=Stephen Gasiorowicz |title=Quantum Physics |year= 1974 |publisher=Wiley & Sons
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| |url=http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471057002.html |isbn=0-471-29281-8 }}
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| </ref><ref name=Merzbacher>
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| {{cite book |author=Eugen Merzbacher |title=Quantum Mechanics |year= 1961 |publisher=Wiley & Sons |isbn=0-471-59670-1 }}
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| </ref> on quantum physics, are often partially replaced by '''cubic harmonics''' for a number of reasons.
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| ==Introduction==
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| The <math>l(l+1)</math> [[Hydrogen-like atom|hydrogen-like atomic orbitals]] with principal quantum number <math>n</math> and angular momentum quantum number <math>l</math> are often expressed as
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| :<math>\psi_{nlm}(\bold{r}) = R_{nl}(r) Y_l^m(\theta,\varphi)</math>
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| in which the <math>R_{nl}(r)</math> is the radial part of the wave function and <math>Y_l^m(\theta,\varphi)</math> is the angular dependent part. The <math>Y_l^m(\theta,\varphi)</math> are the [[spherical harmonics]], which are solutions of the [[angular momentum]] operator. The spherical harmonics are representations of functions of the [[Rotation group SO(3)|full rotation group]] '''SO(3)'''<ref name=BrinkSatchler>
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| {{cite book
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| |author=D. M. Brink & G. R. Satchler
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| |title=Angular Momentum
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| |year= 1993
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| |publisher=Oxford University Press
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| |url=http://www.oup.com/us/catalog/general/subject/Physics/AtomicMolecularOpticalphysics/?view=usa&ci=9780198517597
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| |isbn=0-19-851759-9
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| }}</ref> with rotational symmetry. In many fields of physics and chemistry these spherical harmonics are replaced by '''cubic harmonics''' because the rotational symmetry of the atom and its '''environment''' are '''distorted''' or because cubic harmonics offer '''computational benefits'''.
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| ==Symmetry and coordinate system==
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| In many cases, especially in [[chemistry]] and [[Solid-state physics|solid-state]] and [[condensed-matter physics]], the system under investigation doesn't have rotational symmetry. Often it has some kind of '''lower symmetry''', with a special '''[[Point groups in three dimensions|point group]]''' representation, or it has '''no spatial symmetry at all'''. Biological and [[Biochemistry|biochemical]] systems, like [[amino acid]]s and [[enzyme]]s often belong to low [[molecular symmetry]] point groups. The [[Crystal structure|solid crystals]] of the elements often belong to the [[space group]]s and point groups with high symmetry. (Cubic harmonics representations are often listed and referenced in '''[[List of character tables for chemically important 3D point groups#Character tables|point group tables]]'''.) The system has at least a fixed orientation in three dimensional [[Euclidean space]]. Therefore the coordinate system that is used in such cases is most often a [[Cartesian coordinate system]] instead of a [[spherical coordinate system]]. In a Cartesian coordinate system the [[atomic orbital]]s are often expressed as
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| :<math>\psi_{nlc}(\bold{r}) = R_{nl}(r) X_{lc}(\bold{r})</math> | |
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| with the '''cubic harmonics''',<ref name=McWeeny>
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| {{cite book
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| |author=R. McWeeny
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| |title=Methods of Molecular Quantum Mechanics
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| |year= 1978
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| |publisher=Academic Press
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| |isbn=0-12-486552-6
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| }}</ref><ref name=Muggli>
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| {{cite journal
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| |author=J. Muggli
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| |title=Cubic harmonics as linear combinations of spherical harmonics
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| |publisher=Springer-Verlag
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| |pages=311–317
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| |volume=23
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| |issue=2
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| |journal=Zeitschrift für Angewandte Mathematik und Physik
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| |url=http://www.springerlink.com/content/t07834k2547w15g4/
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| |year=1972
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| |doi=10.1007/BF01593094
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| |bibcode = 1972ZaMP...23..311M }}</ref><ref name=Kwiatkowski>
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| {{cite journal
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| |author=T. Kwiatkowski, S. Olszewski, A. Wierzbicki
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| |title=Cubic harmonics in Cartesian coordinates
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| |pages=21
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| |volume=11
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| |journal=International Journal of Quantum Chemistry
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| |year=1977
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| |doi=10.1002/qua.560110104
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| }}</ref> <math>X_{lc}(\bold{r})</math>, as a [[Basis set (chemistry)|basis set]]. [[LCAO]] and [[Molecular orbital theory|MO]] calculations in [[computational chemistry]] or [[tight binding]] calculations in solid-state physics use cubic harmonics as an atomic orbital basis. The indices ''lc'' are denoting some kind of Cartesian representation.
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| ==Basis transformations==
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| For the [[Orthonormal basis|representations]] of the spherical harmonics a spherical coordinate system is chosen with a '''principal axis''' in the '''z-direction'''. For the cubic harmonics this axis is also the most convenient choice. For states of higher angular momentum quantum number <math>l</math> and a higher dimension of <math>l(l+1)</math> the number of possible rotations or [[Coordinate vector#Basis transformation matrix|basis transformations]] in [[Hilbert space#Quantum mechanics|Hilbert space]] grows and so does the number of possible orthogonal representations that can be constructed on the basis of the <math>l(l+1)</math>-dimensional spherical harmonics basis set. There is more freedom to choose a representation that fits the point group symmetry of the problem. The cubic representations that are listed in [[#Table_of_cubic_harmonics|the table]] are a result of the transformations, which are 45° 2D rotations and a 90° rotation to the real axis if necessary, like
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| :<math>X_{lc}(\bold{r}) = Y_l^0</math> | |
| :<math>X_{lc'}(\bold{r}) = \frac{1}{i^{n_{c'}}\sqrt{2}}\left(Y_l^m - Y_l^{-m}\right)</math>
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| :<math>X_{lc''}(\bold{r}) = \frac{1}{i^{n_{c''}}\sqrt{2}}\left(Y_l^m + Y_l^{-m}\right)</math>
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| A substantial number of the spherical harmonics are listed in the [[Table of spherical harmonics]].
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| ==Computational benefits==
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| [[File:HexacyanidoferratIII 2.svg|thumb|180px|[[Ferricyanide]] ion, used to make 'Turnbull's blue' with an octahedrically surrounded central [[Iron|Fe<sup>3+</sup>]]-ion.]]
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| First of all, the cubic harmonics are ''real functions'', while spherical harmonics are ''[[Complex numbers|complex functions]]''. The complex numbers are two-dimensional with a real part and an imaginary part. Complex numbers offer very handsome and effective tools to tackle mathematical problems analytically but they are not very effective when they are used for numerical calculations. Skipping the imaginary part saves half the calculational effort in summations, a factor of four in multiplications and often factors of eight or even more when it comes to computations involving matrices.
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| The cubic harmonics often fit the symmetry of the potential or surrounding of an atom. A common surrounding of atoms in solids and [[Coordination complex|chemical complexes]] is an octahedral surrounding with an [[Point group character tables#Cubic groups|octahedral cubic point group symmetry]]. The representations of the cubic harmonics often have a high symmetry and multiplicity so operations like integrations can be reduced to a limited, or irreducible, part of the domain of the function that has to be evaluated. A problem with the 48-fold octahedral O<sub>h</sub> symmetry can be calculated much faster if one limits a calculation, like an integration, to the irreducible part of the [[Domain of a function|domain]] of the function.
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| ==Table of cubic harmonics==
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| ===The s-orbitals===
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| The '''s-orbitals''' only have a radial part.
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| :<math>\psi_{n00}(\bold{r}) = R_{n0}(r) Y_0^0</math>
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| :<math>s = X_{00} = Y_0^0 = \frac{1}{\sqrt{4\pi}}</math>
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| {| class="wikitable"
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| |-
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| !
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| ! n=1
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| ! 2
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| ! 3
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| ! 4
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| ! 5
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| ! 6
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| ! 7
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| |-
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| ! R<sub>n0</sub>
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| | [[File:S1M0.png|100px]] || [[File:S2M0.png|100px]] || [[File:S3M0.png|100px]] || [[File:S4M0.png|100px]] || [[File:S5M0.png|100px]] || [[File:S6M0.png|100px]] || [[File:S7M0.png|100px]]
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| |}
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| ===The p-orbitals===
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| The '''three p-orbitals''' are [[atomic orbital]]s with an [[Azimuthal quantum number|angular momentum quantum number]] '''ℓ = 1'''. The cubic harmonic expression of the p-orbitals
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| :<math>p_z = N_1^c \frac{z}{r} = Y_1^0</math>
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| :<math>p_x = N_1^c \frac{x}{r} = -\frac{1}{\sqrt{2}} \left(Y_1^1 - Y_1^{-1}\right)</math>
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| :<math>p_y = N_1^c \frac{y}{r} = i\frac{1}{\sqrt{2}} \left(Y_1^1 + Y_1^{-1}\right)</math>
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| with
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| :<math>N_1^c = \left(\frac{3}{4\pi}\right)^{1/2}</math>
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| {| class="wikitable"
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| |-
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| ! p<sub>z</sub>
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| ! p<sub>x</sub>
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| ! p<sub>y</sub>
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| |-
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| | [[File:P2M0.png|100px]] || [[File:P2M-1.png|100px]] || [[File:P2M1.png|100px]]
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| |}
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| ===The d-orbitals===
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| The '''five d-orbitals''' are [[atomic orbital]]s with an [[Azimuthal quantum number|angular momentum quantum number]] '''ℓ = 2'''. The '''angular part''' of the d-orbitals are often expressed like
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| :<math>\psi_{n2c}(\bold{r}) = R_{n2}(r) X_{2c}(\bold{r})</math>
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| The '''angular part''' of the d-orbitals are the '''cubic harmonics''' <math>X_{2c}(\bold{r})</math>
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| :<math>d_{z^2} = N_2^c \frac{3z^2 - r^2}{2r^2\sqrt{3}} = Y_2^0</math>
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| :<math>d_{xz} = N_2^c \frac{xz}{r^2} = -\frac{1}{\sqrt{2}} \left(Y_2^1 - Y_2^{-1}\right)</math>
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| :<math>d_{yz} = N_2^c \frac{yz}{r^2} = \frac{i}{\sqrt{2}} \left(Y_2^1 + Y_2^{-1}\right)</math>
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| :<math>d_{xy} = N_2^c \frac{xy}{r^2} = -\frac{i}{\sqrt{2}} \left(Y_2^2 - Y_2^{-2}\right)</math>
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| :<math>d_{x^2-y^2} = N_2^c \frac{x^2 - y^2}{2r^2} = \frac{1}{\sqrt{2}} \left(Y_2^2 + Y_2^{-2}\right)</math>
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| with
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| :<math>N_2^c = \left(\frac{15}{4\pi}\right)^{1/2} </math>
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| {| class="wikitable"
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| |-
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| ! d<sub>z<sup>2</sup></sub>
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| ! d<sub>xz</sub>
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| ! d<sub>yz</sub>
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| ! d<sub>xy</sub>
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| ! d<sub>x<sup>2</sup>-y<sup>2</sup></sub>
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| |-
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| | [[File:D3M0.png|100px]] || [[File:D3M-1.png|100px]] || [[File:D3M1.png|100px]] || [[File:D3M-2.png|100px]] || [[File:D3M2.png|100px]]
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| |}
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| ===The f-orbitals===
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| The '''seven f-orbitals''' are [[atomic orbital]]s with an [[Azimuthal quantum number|angular momentum quantum number]] '''ℓ = 3'''. often expressed like
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| :<math>\psi_{n3c}(\bold{r}) = R_{n3}(r) X_{3c}(\bold{r})</math>
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| The '''angular part''' of the f-orbitals are the '''cubic harmonics''' <math>X_{3c}(\bold{r})</math>. In many cases different linear combinations of spherical harmonics are chosen to construct a cubic f-orbital basis set.
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| :<math>f_{z^3} = N_3^c \frac{z (2 z^2 - 3 x^2 - 3 y^2)}{2 r^3 \sqrt{15}} = Y_3^0</math>
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| :<math>f_{xz^2} = N_3^c \frac{x (4 z^2 - x^2 - y^2)}{2 r^3 \sqrt{5}} = \frac{1}{\sqrt{2}}\left(Y_3^1 - Y_3^{-1}\right)</math>
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| :<math>f_{yz^2} = N_3^c \frac{y (4 z^2 - x^2 - y^2)}{r^3 \sqrt{5}} = \frac{1}{i \sqrt{2}}\left(Y_3^1 + Y_3^{-1}\right)</math>
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| :<math>f_{xyz} = N_3^c \frac{xyz}{r^3} = \frac{1}{i \sqrt{2}}\left(Y_3^2 - Y_3^{-2}\right)</math>
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| :<math>f_{z(x^2-y^2)} = N_3^c \frac{z \left( x^2 - y^2 \right)}{2 r^3} = \frac{1}{\sqrt{2}}\left(Y_3^2 + Y_3^{-2}\right)</math>
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| :<math>f_{x(x^2-3y^2)} = N_3^c \frac{x \left( x^2 - 3 y^2 \right)}{2 r^3 \sqrt{3}} = \frac{1}{\sqrt{2}}\left(Y_3^3 - Y_3^{-3}\right)</math>
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| :<math>f_{y(3x^2-y^2)} = N_3^c \frac{y \left( 3 x^2 - y^2 \right)}{2 r^3 \sqrt{3}} = \frac{1}{i \sqrt{2}}\left(Y_3^3 + Y_3^{-3}\right)</math>
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| with
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| :<math>N_3^c = \left(\frac{105}{4\pi}\right)^{1/2} </math>
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| {| class="wikitable"
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| |-
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| ! f<sub>z<sup>3</sup></sub>
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| ! f<sub>xz<sup>2</sup></sub>
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| ! f<sub>yz<sup>2</sup></sub>
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| ! f<sub>xyz</sub>
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| ! f<sub>z(x<sup>2</sup>-y<sup>2</sup>)</sub>
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| ! f<sub>x(x<sup>2</sup>-3y<sup>2</sup>)</sub>
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| ! f<sub>y(3x<sup>2</sup>-y<sup>2</sup>)</sub>
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| |-
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| | [[File:F4M0.png|100px]] || [[File:F4M-1.png|100px]] || [[File:F4M1.png|100px]] || [[File:F4M-2.png|100px]] || [[File:F4M2.png|100px]] || [[File:F4M-3.png|100px]] || [[File:F4M3.png|100px]]
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| |}
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| ==See also==
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| {{multicol}}
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| *[[Atomic orbitals]]
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| *[[Atomic physics]]
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| *[[Spherical harmonics]]
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| *[[Spherical coordinate system]]
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| *[[Cartesian coordinate system]]
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| *[[Euclidean space]]
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| {{multicol-break}}
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| *[[Hilbert space]]
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| *[[Basis set (chemistry)]]
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| *[[Basis (linear algebra)]]
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| *[[Coordinate vector]]
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| *[[Linear combination of atomic orbitals molecular orbital method|LCAO method]]
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| *[[Tight binding|Tight binding method]]
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| {{multicol-end}}
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| ==References==
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| {{reflist|2}}
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| {{atomic models}}
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| [[Category:Molecular physics]]
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| [[Category:Quantum chemistry]]
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| [[Category:Electronic structure methods]]
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| [[Category:Physical chemistry]]
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| [[Category:Euclidean symmetries]]
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